Measure of non-periodicity of almost periodic functions

[reterty]

As is well known, almost periodic functions can be represented as a Fourier series with incommensurable (non-multiple) frequencies https://en.wikipedia.org/wiki/Almost_periodic_function. It seems to me that I came up with an integral criterion for the degree of non-periodicity. The integral of a periodic function (not including the constant component of its Fourier series), with respect to the argument for the main period, is equal to zero. In the theory of almost periodic functions, the concept of an almost period is introduced. So, a similar integral of an almost periodic function for almost a period will be different from zero. Its value divided by this almost period and the largest of the amplitudes of the harmonics of the Fourier series will be a dimensionless quantity characterizing the degree of non-periodicity of this almost periodic function. Is my criterion correct and useful?

 

[DaveE]

The problem with "almost periodic" is that those functions are essentially undefined with such a broad and simple description.

In general, you will have to define over what conditions and how you will do the evaluation. The Fourier transform assumes periodicity based on the limits you choose to integrate over. It can not tell you about any periodicity on the order of 1 day, if you only collect data for 1 minute. So, I think just defining your window and looking at the Fourier transform is the only thing we can do. Then for different circumstances, you'll get different spectral data out, which may still be hard to interpret.

In practice, this subject is most commonly described as "jitter" of electronic signals. It is an extremely well studied and hugely important subject. The treatment tends to be statistical in nature. People invariably end up making some (powerful) assumptions about the type of deviation, like "gaussian noise", for example, to allow them to analyze the more general cases. Do some searching about jitter for more information. IRL, we would look at the spectral width of the "almost periodic" frequency out of a Fourier series or spectrum analyzer. This is also called "phase noise".